Astron. Astrophys. 355, 564-572 (2000)

Dynamical stability for the gravitational evolution of a homogeneous polytrope

C. Ringeval  ^* and S. Bouquet

Commissariat à l'Énergie Atomique, Centre d'étude de Bruyères-le-Châtel, DAM/DIF, Département de Physique Théorique et Appliquée, Service de Physique des Plasmas et Électromagnetisme, 91680 Bruyères-le-Châtel, France

Received 19 July 1999 / Accepted 3 January 2000

Abstract

The dynamic stability of the spherical gravitational evolution (collapse or expansion) for a homogeneous polytropic gas with any exponent , is studied using the lagrangian formalism. We obtain the analytical expression for density perturbations at the first order.

In the case , the Jeans' criterion is easily generalized to a self-similar expanding background. The collapsing case is found to be always unstable. The stability of density modes obtained for does not introduce any conditions on the wavelength perturbation, but only a criterion on the polytropic index. As a result, stability is obtained for an expanding gas provided , and for a collapsing one, for .

Key words: stars: formation hydrodynamics instabilities accretion, accretion disks

* Present adress: DARC, Observatoire de Paris-Meudon, 5, Place Jules Janssen, 92195 Meudon, France

Send offprint requests to: S. Bouquet

Correspondence to: (bouquet@bruyeres.cea.fr; christophe.ringeval@obspm.fr)

This article contains no SIMBAD objects.

Contents

• 1. Introduction
• 2. Eulérian collapse
  • 2.1. Previous results
  • 2.2. Equivalence to the Jeans' criterion
  • 2.3. Conclusion
• 3. Lagrangian collapse
  • 3.1. Equation of evolution
  • 3.2. Density eigenmodes
    • 3.2.1. Mass dependence of the solution
    • 3.2.2. Time dependence of the solution
  • 3.3. Stability of eigenmodes
    • 3.3.1. Stability of the eigenmodes for
    • 3.3.2. Stability of eigenmodes with
  • 3.4. Perturbations in a finite medium and influence of boundary conditions
    • 3.4.1. Case
    • 3.4.2. Case
• 4. Conclusion
• Acknowledgements
• References

© European Southern Observatory (ESO) 2000

Online publication: March 9, 2000
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